Question

A spherical snowball is melting such that its volume is decreasing at the rate of \(1\ \text{cm}^3/\text{min}\). The rate at which the diameter is decreasing when the diameter is 10 cm is

• A. \( \frac{11}{75\pi} \) cm/min
• B. \( \frac{1}{50\pi} \) cm/min
• C. \( \frac{2}{75\pi} \) cm/min
• D. \( \frac{1}{25\pi} \) cm/min

Hint:

In related rates, always express all variables in terms of one variable before differentiating.

Answer 1

The Correct Option is B

Step 1: Write the volume formula.
Volume of a sphere is:
\[ V = \frac{4}{3}\pi r^3. \]

Step 2: Differentiate with respect to time.
\[ \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}. \]

Step 3: Relate radius and diameter.
\[ r = \frac{D}{2} \Rightarrow \frac{dr}{dt} = \frac{1}{2}\frac{dD}{dt}. \]

Step 4: Substitute in derivative.
\[ \frac{dV}{dt} = 4\pi r^2 \cdot \frac{1}{2}\frac{dD}{dt}. \]
\[ \frac{dV}{dt} = 2\pi r^2 \frac{dD}{dt}. \]

Step 5: Substitute given values.
Given:
\[ \frac{dV}{dt} = -1,\quad D=10 \Rightarrow r=5. \]
\[ -1 = 2\pi (5)^2 \frac{dD}{dt}. \]
\[ -1 = 50\pi \frac{dD}{dt}. \]

Step 6: Solve for \( \frac{dD}{dt} \).
\[ \frac{dD}{dt} = -\frac{1}{50\pi}. \]

Step 7: Interpret result.
Since diameter is decreasing, magnitude is:
\[ \frac{1}{50\pi}. \]
Final Answer:
\[ \boxed{\frac{1}{50\pi}} \]